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We are told:
#sf((dX)/dt=8color(white)(x)"ft/s")#
We need to find #sf(("d"theta)/dt#
From the geometry of the situation, Pythagoras tells us:
#sf(X^2+100^2=200^2)#
#sf(X^2+10,000=40,000)#
#sf(X=sqrt(30,000)color(white)(x)ft)#
and
#sf(sintheta=100/200=0.5)#
#sf(theta=30^@)#
At this instant:
#sf(tantheta=100/X)#
Differentiating implicitly with respect to time t :
#sf(d/dt[tantheta]=d/dt[100/X])#
#sf(sec^2theta.("d"theta)/dt=-100/(X^2).(dX)/dt)#
#sf(("d"theta)/dt=-(100cos^2theta)/(X^2).(dX)/dt)#
Putting in the numbers:
#sf(("d"theta)/dt=-100xxcos^2(30)/(30,000)xx8)#
#sf(("d"theta)/dt=-100xx0.75/(30,000)xx8=-0.02color(white)(x)"rad/s")#
#sf(2picolor(white)(x)"rad"=360^@)#
#:.##sf(1color(white)(x)"rad"=360/(2pi)=57.3^@)#
#:.##sf(0.02color(white)(x)"rad"=57.2xx0.02=1.15^@)#
In degrees:
#sf(("d"theta)/dt=-1.15^@"/s")#
